A Generalized Age-Dependent Branching Process and Its Limit Distribution | ||
| The Egyptian Statistical Journal | ||
| Article 2, Volume 14, Issue 1, June 1970, Pages 13-31 PDF (11.33 M) | ||
| Document Type: Original Article | ||
| DOI: 10.21608/esju.1970.315803 | ||
| Author | ||
| I. N. Shimi | ||
| Abstract | ||
| An age-dependent continuous parameter branching stochastic process Xₙ(t) is considered, where Xₙ (t) is the number of particles in the population at time t and N is the initial size of the population. If the probabilities of splits depend on N. in a way that will be made precise in the text, then, for fixed t. as N tends to infinity a limiting distribution of the stochastic processes Xₙ (t)__N is obtained, and shown to be the distribution of a continuous parameter stochastic process with independent increments X (t), whose distribution will be determined. To establish this we need to prove that for t₁< t₂ <... <tₙ the distribution of Xₙ (tᵢ) -- X(tᵢ₋₁) converges to the distribution of X(tᵢ) --X(tᵢ₋₁), and Xₙ (t₂) -- Xₙ (t₁). Xₙ(t₃) -- Xₙ(t₂),…., Xₙ (tₙ) -- Xₙ ( tₙ₋₁) are independent in the limit. The limiting distribution of Xₙ (t) - N gives an approximation to the distribution of Xₙ (t) for any fixed t and large N. | ||
| Keywords | ||
| Branching Process; Stochastic Process; Generalized Distribution | ||
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